Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 156901, 4 pages
http://dx.doi.org/10.1155/2013/156901
Research Article
The Capacity Expansion Path Problem in Networks
Jianping Li and Juanping Zhu
Department of Mathematics, Yunnan University, Kunming 650091, China
Correspondence should be addressed to Juanping Zhu; juanpingzhu@gmail.com
Received 8 February 2013; Revised 1 June 2013; Accepted 25 June 2013
Academic Editor: Yuri Sotskov
Copyright © 2013 J. Li and J. Zhu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper considers the general capacity expansion path problem (GCEP) for the telecommunication operators. We investigate
the polynomial equivalence between the GCEP problem and the constrained shortest path problem (CSP) and present a
pseudopolynomial algorithm for the GCEP problem, no matter the graph is acyclic or not. Furthermore, we investigate two special
versions of the GCEP problem. For the minimum number arc capacity expansion path problem (MN-CEP), we give a strongly
polynomial algorithm based on the dynamic programming. For the minimum-cost capacity expansion shortest path problem
(MCESP), we give a strongly polynomial algorithm by constructing a shortest paths network.
1. Introduction
Currently Chinese telecommunication operators are under
the huge pressure of communication capacity and communication quality, followed by the rapidly increasing users.
Shortest path problem plays an important role in the telecommunication networks. In this paper, we consider the network
capacity expansion strategy for the telecommunication operators, called the general capacity expansion path problem
(GCEP), given a weighted digraph 𝐷 = (𝑉, 𝐴; 𝑤, 𝑐, 𝑝) with
𝑛 nodes, 𝑚 arcs, and two fixed nodes 𝑠, 𝑡. For each arc 𝑒 ∈ 𝐴,
nonnegative 𝑤(𝑒) is the weight, 𝑐(𝑒) is the capacity, and nonnegative 𝑝(𝑒) is the cost of expanding the capacity of the arc 𝑒
by one unit. Given two positive 𝑊 and 𝑑, find a path 𝑃𝑠,𝑡 from
𝑠 to 𝑡 to satisfy two constraints: (1) 𝑤(𝑃𝑠,𝑡 ) = ∑𝑒∈𝑃𝑠,𝑡 𝑤(𝑒) ≤ 𝑊
and (2) for some arc 𝑒 ∈ 𝐴, if the capacity 𝑐(𝑒) < 𝑑, we
must expand the capacity of arc 𝑒 to 𝑑. The objective is to
minimize the total expansion cost ∑𝑒∈𝑃𝑠,𝑡 add(𝑒)𝑝(𝑒) on the
path 𝑃𝑠,𝑡 , where add(𝑒) = 𝑑 − 𝑐(𝑒) if 𝑐(𝑒) < 𝑑 and add(𝑒) = 0
otherwise. In telecommunication network, 𝑊 is the given
maximum length of the path 𝑃𝑠,𝑡 , and 𝑑 is the expected
maximum number of the users. If the length of the path 𝑃𝑠,𝑡
exceeds 𝑊, the communication quality would drop quickly.
If the capacity (or bandwidth) of the arc is not wide enough,
the arc would be congested, and we have to expand this arc to
the expected 𝑑.
Krumke et al. [1] studied budget constrained network
upgrading problems. The goal is how to find a minimum cost
set of vertices to be upgraded so that the resulting network has
a minimum spanning tree of weight no more than the budget.
Their upgrading costs are on the vertices.
Zhang et al. [2] studied the problem how to increase
the capacities of the elements in an edge set 𝐸 efficiently so
that the capacity of a given family F of subsets of 𝐸 can be
increased to the maximum extent, where the total cost for
the increment of capacity is within a given budget bound 𝐷.
Yang and Zhang [3] considered a class of bottleneck capacity
expansion problems, which aimed to enhance bottleneck
capacity to a certain level with minimum cost. There were
two types of expanding models: arc expanding and node
expanding. They emphasized the maximum capacity path
problem. Our models are different from theirs. For example,
when we consider the arc expansion cost of the path in the
telecommunication network, on which each signal is sent
from the source 𝑠 via some transferring facility V𝑖 to the sink
𝑡, we must make sure that the transferring length does not
exceed the length bound 𝑊.
Different from our model and method, Seref et al. [4]
studied an incremental optimization problem. They started
from a feasible solution 𝑥0 and made an incremental change
in 𝑥0 that would result in the greatest improvement in
the objective function. For the incremental shortest path
2
problem, they showed that the arc inclusion version is polynomially solvable whereas the arc exclusion version is 𝑁𝑃complete.
In this paper, we present a proof of polynomial equivalence between the GCEP problem and the constrained
shortest path problem (CSP), which leads to that the GCEP
problem remains 𝑁𝑃-hard. We design a pseudopolynomial
algorithm and a FPTAS to solve the GCEP problem. Then
we study two special versions of the GCEP problem. For
the first version, minimum number arc capacity expansion
path problem (MNCEP), we present a strongly polynomial
algorithm based on dynamic programming. For the second
version, the minimum-cost capacity expansion shortest path
problem (MCESP), we construct a shortest path network and
present a strongly polynomial algorithm.
2. The Equivalence between the GCEP
Problem and the CSP Problem
Motivated by the polynomial equivalence between the Hitchcock problem and the minimum-cost flow problem [5], we
prove that the GCEP problem is polynomially equivalent
to the constrained shortest path problem, which leads to
Theorem 1. Hassin [6] studied the constrained (or restricted)
shortest path problem (CSP) as follows. Let 𝐷 = (𝑉; 𝐴) be a
weighted digraph with 𝑛 nodes, 𝑚 arcs, and two fixed nodes
𝑠, 𝑡. Each arc 𝑒 ∈ 𝐴 has a length 𝑐(𝑒) and a transition time
𝑤(𝑒). These numbers are assumed to be positive integers.
For any path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 in 𝐷, the length 𝑐(𝑃𝑠,𝑡 ) and
transition time 𝑤(𝑃𝑠,𝑡 ) are defined as the sum of the lengths
and transition times of all arcs on 𝑃𝑠,𝑡 , respectively. The
CSP problem is to find the minimum length path 𝑃𝑠,𝑡 in 𝐷
such that transition time 𝑤(𝑃𝑠,𝑡 ) along this path does not
exceed a given bound 𝑊. For convenience, we denote a
notation, by {𝐷 = (𝑉, 𝐴; 𝑤, 𝑐; 𝑠, 𝑡); 𝑊}CSP , to represent an
instance of the CSP problem. The CSP problem is a classical 𝑁𝑃-hard problem [7]. To obtain Theorem 1, we use
{𝐷 = (𝑉, 𝐴; 𝑤, 𝑐, 𝑝; 𝑠, 𝑡); 𝑊; 𝑑}GCEP to represent an instance of
the GCEP problem.
Theorem 1. The GCEP problem is polynomially equivalent to
the CSP problem.
Proof. It is sufficient to prove that the GCEP problem can be
transformed to the CSP problem in polynomial operations,
and vice versa.
For any instance I, say {𝐷 = (𝑉, 𝐴; 𝑤, 𝑐, 𝑝; 𝑠, 𝑡);
𝑊; 𝑑}GCEP , of the GCEP problem, we construct an instance
𝜏(I) of the CSP problem: a digraph {𝐷󸀠 = (𝑉, 𝐴; 𝑤󸀠 , 𝑐󸀠 ; 𝑠, 𝑡);
𝑊󸀠 }CSP consisting of the same structure as the digraph 𝐷
with 𝑛 nodes, 𝑚 arcs, two fixed nodes 𝑠 and 𝑡, and two
positive 𝑑 and 𝑊󸀠 (= 𝑊). For each arc 𝑒 ∈ 𝐴, define two
nonnegative 𝑤󸀠 (𝑒) = 𝑤(𝑒) and 𝑐󸀠 (𝑒) = add(𝑒)𝑝(𝑒), where
add(𝑒) = 𝑑 − 𝑐(𝑒) if 𝑐(𝑒) < 𝑑 and add(𝑒) = 0 otherwise. The
objective is to find a path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 in 𝐷󸀠 such that
∑𝑒∈𝑃𝑠,𝑡 𝑤󸀠 (𝑒) ≤ 𝑊󸀠 and the value of ∑𝑒∈𝑃𝑠,𝑡 𝑐󸀠 (𝑒) is minimized.
Journal of Applied Mathematics
On the converse direction, for any instance J, say
{𝐷󸀠 = (𝑉, 𝐴; 𝑤󸀠 , 𝑐󸀠 ; 𝑠, 𝑡); 𝑊󸀠 }CSP , of the CSP problem, we construct an instance 𝛼(J) of the GCEP problem: a digraph
𝐷 = (𝑉, 𝐴; 𝑤, 𝑐, 𝑝; 𝑠, 𝑡; 𝑊, 𝑑)GCEP consisting of the same structure as the digraph 𝐷󸀠 with 𝑛 nodes, 𝑚 arcs, a fixed positive
integer 𝑑 = 2, and the bound 𝑊 = 𝑊󸀠 . For each arc 𝑒 ∈ 𝐴,
define nonnegative 𝑤(𝑒) = 𝑤󸀠 (𝑒), 𝑐(𝑒) = 1, and 𝑝(𝑒) = 𝑐󸀠 (𝑒),
which implies that add(𝑒) = 1 and add(𝑒)𝑝(𝑒) = 𝑐󸀠 (𝑒). The
objective is to find a path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 in 𝐷󸀠 such that
∑𝑒∈𝑃𝑠,𝑡 𝑤(𝑒) ≤ 𝑊(= 𝑊󸀠 ) and the value of ∑𝑒∈𝑃𝑠,𝑡 add(𝑒)𝑝(𝑒) =
∑𝑒∈𝑃𝑠,𝑡 𝑐󸀠 (𝑒) is minimized.
It is easy to prove the claim: there is an optimal solution to
the instance J of the CSP problem with the optimal value 𝑘 if
and only if there is an optimal solution to the instance 𝛼(J)
of the GCEP problem with the optimal value 𝑘. Moreover, the
transformation is executed in polynomial operations.
Since the CSP problem is NP-hard, the proof of
Theorem 1 implies that the GCEP problem is NP-hard, even
if 𝑐(𝑒) = 1. And if 𝑝(𝑒) = 1, the GCEP problem is reduced to
regular CSP problem.
3. A Pseudopolynomial Algorithm for
the GCEP Problem
According to Theorem 1, for any instance I of the GCEP
problem in the acyclic digraphs, by constructing an instance
𝜏(I) of the CSP problem and utilizing the FPTAS in [6, 8]
for the CSP problem, we can get a FPTAS to solve the GCEP
problem. Now for the GCEP problem, no matter acyclic
digraphs or not, by applying the following dynamic programming method which is similar to the Bellman-Ford algorithm
[9, 10], we give a pseudopolynomial algorithm. To describe
our algorithm, for each node 𝑢 ∈ 𝑉 and two integers 𝑘 ≥ 0
and 𝑔 ≥ 0, define 𝑓𝑘 (𝑢, 𝑔) to be the minimum length of an
optimal path 𝑃𝑠,𝑢 from 𝑠 to 𝑢 with respect to the weight 𝑤,
such that at most 𝑘 arcs are traversed and the expansion cost
along 𝑃𝑠,𝑢 is at most 𝑔. Set 𝑓𝑘 (𝑢, 𝑔) := +∞ if no such path
exists. For convenience, denote addcost(𝑒) = add(𝑒)𝑝(𝑒) as
the expansion cost for each arc 𝑒 ∈ 𝐴 and 𝑊-path as the
directed path if its weight is less than or equal to 𝑊. Set OPT
to be the minimum capacity expansion cost of 𝑠-𝑡 𝑊-path.
Algorithm 2 (EXACT(𝑊)). Comment. Return minimum
capacity expansion path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 with constraint
𝑤(𝑃𝑠,𝑡 ) ≤ 𝑊.
Begin
Step 1 (initialization). Set 𝑓0 (𝑠, 𝑔) = 0 for each 𝑔 ≥ 0 and
set 𝑓0 (𝑢, 𝑔) = +∞ if 𝑢 ≠ 𝑠.
Step 2
For 𝑔 = 1, 2, . . . , OPT, do
For 𝑘 = 1, 2, . . . , 𝑛 − 1, do
For each 𝑢 ∈ 𝑉, do
Journal of Applied Mathematics
𝑓𝑘 (𝑢, 𝑔) = min{𝑓𝑘−1 (𝑢, 𝑔), min{V|(V,𝑢)∈𝐴,addcost(V,𝑢)≤𝑔}
{𝑓𝑘−1 (V, 𝑔 − addcost(V, 𝑢)) + 𝑤(V, 𝑢)}};
If 𝑓𝑛−1 (𝑡, 𝑔) ≤ 𝑊, output OPT = 𝑔 and the path 𝑃𝑠,𝑡 ,
and exit.
End of Algorithm EXACT(𝑊).
Actually algorithm EXACT(𝑊) is a dynamic programming algorithm, which generalizes the Bellman-Ford method
[9, 10]. Note that OPT is not known but it satisfies OPT =
min{𝑔 | 𝑓𝑛−1 (𝑡, 𝑔) ≤ 𝑊}. 𝑘 is the maximum number of
traversing arcs on the path from node 𝑠 to 𝑢, and the
algorithm enumerates each 𝑘(≤ 𝑛 − 1) and each 𝑢 ∈ 𝑉 for
𝑓𝑘 (𝑢, 𝑔). To get 𝑓𝑛−1 (𝑡, 𝑔), the algorithm computes 𝑓𝑘 (𝑢, 𝑔)
first for 𝑔 = 1, 𝑘 = 1, . . . , 𝑛 − 1, and each 𝑢 ∈ 𝑉, then for
𝑔 = 2, 𝑘 = 1, . . . , 𝑛 − 1, and each 𝑢 ∈ 𝑉 and so on, until
the first value of 𝑔 for which 𝑓𝑛−1 (𝑡, 𝑔) ≤ 𝑊. According to
the definition of 𝑓𝑘 (𝑢, 𝑔), 𝑓𝑘 (𝑡, 𝑔) is the minimum length of
an optimal path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 with respect to the weight 𝑤,
such that at most 𝑘 arcs are traversed and the expansion cost
along 𝑃𝑠,𝑡 is at most 𝑔. 𝑓𝑛−1 (𝑡, 𝑔) ≤ 𝑊 indicates the length
with respect to weight 𝑤 on this optimal path 𝑃𝑠,𝑡 is less than
and equal to 𝑊 (optimal condition), and 𝑔 ≤ OPT. Hence
OPT is set to this value of 𝑔. So the complexity of algorithm
EXACT(𝑊) is O(OPT ⋅ 𝑚𝑛2 ), and we obtain Theorem 3.
Theorem 3. For the GCEP problem, the algorithm EXACT(𝑊)
produces an optimal solution, and its time complexity is
O(OPT ⋅ 𝑚𝑛2 ).
Moreover, the technique of rounding and scaling for
acyclic digraphs in [8] is applied to our pseudopolynomial
algorithm EXACT(𝑊), and we can present an FPTAS in time
O(𝑛2 𝑚/𝜀) for the GCEP problem even in general digraphs,
where 𝜀 > 0. The construction of our FPTAS for general
digraphs is similar to the one due to [8] for acyclic digraphs.
The reader can find the algorithm in details from [8]; thus we
omit it here.
4. Dynamic Programming Algorithm for
the MNCEP Problem
In this section, we study the first special version of the
GCEP problem, minimum number arc capacity expansion
path problem (MNCEP), where for each arc 𝑒 ∈ 𝐴, 𝑝(𝑒) = 1,
add(𝑒) = 1 if 𝑐(𝑒) < 𝑑 and add(𝑒) = 0 otherwise. Equivalently,
the objective is to find a path 𝑃𝑠,𝑡 of 𝐷, on which the number
of arcs to be expanded is minimized.
Like the GCEP problem, no matter acyclic digraphs
or not, by applying the method similar to Bellman-Ford
algorithm [9, 10], we can design a strongly polynomial
algorithm to solve the MNCEP problem. For each node 𝑢 ∈
𝑉, given two integers 𝑖 ≥ 0 and 𝑗 ≥ 0, define 𝑓(𝑢, 𝑖, 𝑗) to be the
minimum length of the path 𝑃𝑠,𝑢 from 𝑠 to 𝑢 with respect to
the weight 𝑤, which traverses at most 𝑖 arcs with exact 𝑗 arcs
to be expanded. Set 𝑓(𝑢, 𝑖, 𝑗) := +∞ if no such path exists.
3
Algorithm 4 (MNCEP). Comment. Return the minimum
number of arcs to be expanded on the path 𝑃𝑠,𝑡 from 𝑠 to 𝑡
with constraint 𝑤(𝑃𝑠,𝑡 ) ≤ 𝑊.
Begin
Step 1 (initialization). Set 𝑓(𝑠, 𝑖, 𝑗) = 0 and set 𝑓(𝑢, 𝑖, 𝑗) =
+∞ if 𝑢 ≠ 𝑠.
Step 2
For 𝑖 = 1, 2, . . . , 𝑛 − 1, do
For 𝑗 = 1, 2, . . . , 𝑖, do
For each node 𝑢
∈
𝑉, do 𝑓(𝑢, 𝑖, 𝑗)
=
min{𝑓(𝑢, 𝑖 − 1, 𝑗), min{V|(V,𝑢)∈𝐴,𝑐(V,𝑢)≥𝑑} {𝑓(V, 𝑖 − 1, 𝑗) +
𝑤(V, 𝑢)},min{V|(V,𝑢)∈𝐴,𝑐(V,𝑢)<𝑑} {𝑑(V, 𝑖−1, 𝑗−1)+𝑤(V, 𝑢)}}
If 𝑓(𝑡, 𝑖, 𝑗) ≤ 𝑊, then output OPT = 𝑗 and the path
𝑃𝑠,𝑡 , and exit.
End of Algorithm MNCEP.
The algorithm MNCEP is also based on dynamic
programming. Different from algorithm EXACT(𝑊), no
unknown OPT appears in the algorithm. 𝑖 is the maximum
number of traversing arcs on the path from node 𝑠 to 𝑢, and
𝑗 is the exact number of arcs to be expanded on this path. So
𝑗 ≤ 𝑖. The algorithm enumerates each 𝑖(≤ 𝑛 − 1), each 𝑗 ≤ 𝑖,
and each 𝑢 ∈ 𝑉 till 𝑓(𝑡, 𝑖, 𝑗) ≤ 𝑊. So the algorithm computes
𝑓(𝑢, 𝑖, 𝑗) first for 𝑖 = 1, 𝑗 = 1, 2, . . . , 𝑖, and each 𝑢 ∈ 𝑉, then
for 𝑖 = 2, 𝑗 = 1, 2, . . . , 𝑖, and each 𝑢 ∈ 𝑉, and so on, until
the first value 𝑖 ≤ 𝑛 − 1 for which 𝑓(𝑡, 𝑖, 𝑗) ≤ 𝑊. Hence,
output OPT = 𝑗. Thus, the algorithm runs in O(𝑚𝑛2 ) time.
We obtain Theorem 5.
Theorem 5. For the MNCEP problem, the algorithm MNCEP
produces an optimal solution, and its time complexity is
O(𝑚𝑛2 ).
5. Strongly Polynomial Algorithm for
the MCESP Problem
The second special version of GCEP is the minimumcost capacity expansion shortest path problem (MCESP). In
MCESP problem, the bound 𝑊 is set to be the shortest
distance dist(𝑠, 𝑡) from 𝑠 to 𝑡 in 𝐷, with respect to weight
𝑤. So the objective of MCESP problem is to find a shortest
path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 with respect to the weight 𝑤, such
that the total expansion cost ∑𝑒∈𝑃𝑠,𝑡 add(𝑒)𝑝(𝑒) on the path
𝑃𝑠,𝑡 is minimized, where add(𝑒) = 𝑑 − 𝑐(𝑒) if 𝑐(𝑒) < 𝑑
and add(𝑒) = 0 otherwise. To solve this MCESP problem
optimally, the shortest paths network plays an important role
in the algorithm.
Property 1 (see [11]). Let the vector dist represent the shortest
paths distances. Then a directed path 𝑃𝑠,V from the source
node 𝑠 to the node V is a shortest path if and only if dist(𝑠, V) =
dist(𝑠, 𝑢) + 𝑤(𝑢, V) for every arc (𝑢, V) ∈ 𝑃𝑠,V .
For convenience, set addcost(𝑒) = add(𝑒) ⋅ 𝑐(𝑒) as the
capacity expansion cost for each arc 𝑒 ∈ 𝐴.
4
Algorithm 6 (MCESP). Comment. Return the minimum-cost
capacity expansion shortest path.
Begin
Step 1. Perform the breadth-first search in the digraph
𝐷, using the arcs to satisfy the equality dist(𝑠, V) =
dist(𝑠, 𝑢) + 𝑤(𝑢, V). Construct the shortest paths network
𝐷󸀠 = (𝑉, 𝐴󸀠 ; 𝑤, addcost) of 𝐷, where 𝐴󸀠 = {(𝑢, V) ∈ 𝐴 |
dist(𝑠, V) = dist(𝑠, 𝑢) + 𝑤(𝑢, V)}. The values of 𝑤 and addcost
in 𝐷󸀠 remain the same as in 𝐷.
Step 2. Find the shortest path 𝑃𝑠,𝑡 from 𝑠 to 𝑡 in digraph
𝐷󸀠 , with respect to the arc capacity expansion cost addcost.
Output OPT = addcost (𝑃𝑠,𝑡 ) and the path 𝑃𝑠,𝑡 .
End of Algorithm MCESP.
Theorem 7. The algorithm MCESP optimally solves the
MCESP problem, and it runs in the time O(𝑛2 ).
Proof. Property 1 implies the fact: if 𝑃𝑠,V is a shortest path from
the source 𝑠 to some node V, then dist(𝑠, V) = dist(𝑠, 𝑢) +
𝑤(𝑢, V) for every arc (𝑢, V) ∈ 𝑃𝑠,V . The converse is also true: if
dist(𝑠, V) = dist(𝑠, 𝑢) + 𝑤(𝑢, V) for every arc (𝑢, V) in a directed
path 𝑃𝑠,V from the source 𝑠 to the node V, then 𝑃𝑠,V must be a
shortest path. The algorithm MCESP searches all 𝑚 arcs for
dist(𝑠, V) = dist(𝑠, 𝑢) + 𝑤(𝑢, V) in 𝐷. Hence, the new digraph
𝐷󸀠 includes all minimum distance path from 𝑠 to 𝑡 in 𝐷, with
respect to the weight 𝑤. The Step 2 of the algorithm selects
the minimum cost capacity expansion path 𝑃𝑠,𝑡 in 𝐷󸀠 . Hence,
the algorithm MCESP solves the MCESP problem optimally.
Step 1 of the algorithm needs O(𝑛2 ) steps to compute all
𝑑(𝑠, V) by using Dijkstra algorithm for shortest paths. Step
2 needs O(𝑚) steps to perform breadth-first search. Step 2
needs O(𝑛2 ) steps as Step 1. Hence, the time complexity of the
algorithm MCESP is O(𝑛2 ).
6. Conclusion
In this paper, we focus on the general capacity expansion
path problem for telecommunication operators, a new practical capacity expansion model. The polynomial equivalence
between the GCEP problem and the CSP problem is proved.
Then by utilizing dynamic programming technique and
Bellman-Ford method [9, 10], we design a pseudopolynomial
algorithm for the GCEP problem. We also present two special
versions of the GCEP problem: the MNCEP problem and
the MCESP problem. For the MNCEP problem, by using
dynamic programming method and Bellman-Ford method,
we design a strongly polynomial algorithm to solve it. For the
MCESP problem, by constructing a shortest paths network,
we design a strongly polynomial algorithm.
For further work, we would investigate some other special
versions of the GCEP problem and related algorithms, which
exist in the current telecommunication networks.
Acknowledgment
The work is fully supported by the National Natural Science
Foundation of China [no. 11126355, 61063011].
Journal of Applied Mathematics
References
[1] S. O. Krumke, M. V. Marathe, H. Noltemeier et al., “Improving
minimum cost spanning trees by upgrading nodes,” Journal of
Algorithms, vol. 33, no. 1, pp. 92–111, 1999.
[2] J. Zhang, C. Yang, and Y. X. Lin, “A class of bottleneck expansion
problems,” Computers and Operations Research, vol. 28, no. 6,
pp. 505–519, 2001.
[3] C. Yang and J. Z. Zhang, “On the bottleneck capacity expansion
problem on networks,” Acta Mathematica Scientia, vol. 26, no.
2, pp. 202–208, 2006.
[4] O. Seref, R. K. Ahuja, and J. B. Orlin, “Incremental network optimization: theory and algorithms,” Operations Research, vol. 57,
pp. 586–594, 2009.
[5] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prince Hall, 1982.
[6] R. Hassin, “Approximation schemes for the restricted shortest
path problem,” Mathematics of Operations Research, vol. 17, no.
1, pp. 36–42, 1992.
[7] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman,
San Francisco, Calif, USA, 1979.
[8] F. Ergun, R. Sinha, and L. Zhang, “An improved FPTAS for
restricted shortest path,” Information Processing Letters, vol. 83,
no. 5, pp. 287–291, 2002.
[9] R. Bellman, “On a routing problem,” Quarterly of Applied Mathematics, vol. 16, pp. 87–90, 1958.
[10] L. R. Ford, “Network flow theory,” The RAND Corporation
Paper, p. 923, 1956.
[11] R. K. Ahuja, L. Magnanti, and B. Orlin, Network Flows: Theory,
Algorithms, and Applications, Printice Hall, 1993.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014